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International Journal of Impact Engineering 34 (2007) 647–667

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Impact of aircraft rubber tyre fragments on aluminium alloyplates: II—Numerical simulation using LS-DYNA

D. Karagiozovaa, R.A.W. Minesb,�

aInstitute of Mechanics, Bulgarian Academy of Sciences, Acad. G. Bonchev Street, Block 4, Sofia 1113, BulgariabImpact Research Centre, University of Liverpool, Brownlow Street, Liverpool, L69 3GH, England

Received 16 November 2005; received in revised form 13 February 2006; accepted 17 February 2006

Available online 8 June 2006

Abstract

A discrete model for a reinforced rubber-like material is proposed in order to simulate numerically a debris tyre impact

on a typical structure of an aircraft when using the FE code LS-DYNA. The model is calibrated using the static and

dynamic test data for the actual tyre material.

The dynamics of the tyre projectile is validated when comparing the numerical predictions with the response of a square

aluminium alloy plate subject to a ribbon projectile impact having different initial velocities and impacting the plate at an

angle of 301. Good agreement is obtained in terms of the strains in the plate caused by the ribbon impact. The numerically

predicted deformations of the projectile also represent well the dynamics of the tyre ribbon recorded during the

experiments. Some characteristic features of a soft projectile impact on a deformable plate are discussed.

The developed model is then extended to the simulation of a full-scale impact test, and good agreement is shown between

experiment and simulation.

r 2006 Elsevier Ltd. All rights reserved.

Keywords: Reinforced rubber; Discrete material model; Soft impact; Numerical simulation

1. Introduction

The rebound of a projectile from a surface has been intensively studied in connection with the ballisticmechanics of ricochet for rigid and deformable bodies [1,2]. A new insight into the response of structuralcomponents associated with elastic–plastic large deformations is provided nowadays by the numericalsimulations (e.g. [3–5]), aiming at an improvement of design and of an increase in product safety.

The vast majority of studies on projectile rebound from a structure consider both the structure and theprojectile to have a comparable stiffness and hence the subsequent recovery of the stored elastic energy causesmotion changes in the projectile. In the case of an impact of a rubber like projectile on a metal plate, asignificant proportion of the initial kinetic energy participates in the deformation of the projectile, so that thekinetic energy transferred to the plate depends significantly on the interaction between the plate and the

ee front matter r 2006 Elsevier Ltd. All rights reserved.

mpeng.2006.02.004

ing author. Tel.: +44151 794 4819; fax: +44151 794 4848.

ess: [email protected] (R.A.W. Mines).

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dx.doi.org/10.1016/j.ijimpeng.2006.02.004

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Nomenclature

A cross-sectional area of the fibresA1, A2 material parameters for the Mooney–Rivlin law (rubber part), Eq. (1)Cij contravariant strain tensorD material constantsdy distance between reinforcing cords in the y directionEc elastic modulus of the fibre materialE elastic modulus of the aluminium alloyfz number of cord layers per unit thicknessIi strain invariantsJel elastic volume ratioJ total volume ratioK0 bulk modulust true stressTk kinetic energy (plate or tyre projectile)Tc deformation energy (plate or tyre projectile)W strain energy potentialb scaling coefficient, Eq. (6)dij covariant strain tensorli deviatoric extension ratiosm0 shear modulusnr, nc the Poisson ratios for the rubber part and reinforcing fibres, respectivelyr0 actual density of the tyre materialrr, rc density of the rubber part and reinforcing fibres, respectivelys nominal stress

D. Karagiozova, R.A.W. Mines / International Journal of Impact Engineering 34 (2007) 647–667648

projectile, and on the dynamics of the projectile during impact. Different angles of impact result in differentprojectile trajectories, which also contribute to the variation of the forces and energy transmitted to the plate.

The aircraft tyre materials used in practice have a complex structure consisting of a rubber-like part andreinforcement [6]. The reinforcing cords are ropes twisted from Nylon fibre and their stiffness in compressionis significantly lower then the stiffness in tension, moreover they are placed at different angles [6,7]. Thismaterial structure causes anisotropic properties in the different directions of loading and non-linear stress-strain dependence when subjected to large deformations. The complex material structure requires an adequatemodelling, which can represent correctly the stiffness of the impactor since this characteristic can significantlyinfluence the interaction with a structure due to the large flexibility of the soft projectile.

Several approaches are usually used in the finite element analysis of cord-reinforced composites. One classof computational models are based on the analysis techniques for laminated fibrous composites [8,9]. Thematerial properties of the individual components are averaged over a layer of a laminated shell element anddescribed by an anisotropic material law. This approach however, requires assumptions for a number ofmaterial constants related to the ‘averaged’ layers, which causes difficulties prior to the analysis and after, forthe recovering of the properties of the individual components. This approach could have restricted applicationin the case of very large anisotropy, which might result in a non-positive defined elasticity matrix.

An alternative approach is to use one-dimensional bar elements to simulate the behaviour of the reinforcingcords. These elements are superimposed on the rubber elements by satisfying the compatibility conditions attheir common nodes. This approach was first proposed for reinforced concrete [10] and applied later to rubbercomposites [7,11]. The concept is simple and allows the modelling of anisotropic material properties ofreinforced rubber when varying the bar position and orientation of the bar elements. The major disadvantageof this approach is that the locations and orientation of the cords are restricted by the finite elementdiscretisation, which could present difficulties for a high number of reinforcing cords. However, the low

ARTICLE IN PRESSD. Karagiozova, R.A.W. Mines / International Journal of Impact Engineering 34 (2007) 647–667 649

computational cost and relatively small number of the required material constants make this approachattractive, particularly when the material parameters need to be obtained by calibration.

Another efficient approach has been developed to model cord-reinforced composites, which is characterisedby superimposing so-called ‘rebar’ elements, consisting of one or more reinforced cord layers with arbitraryorientations, on corresponding rubber elements. This technique was originally applied for reinforced concrete[12] and is widely used nowadays to model cord reinforced rubber [13–16]. Two and three dimensional rebarelements have been implemented in FE codes such as MARC and ABAQUS. This approach is discretisationindependent and is probably the most effective for modelling reinforced composites.

A further step in modelling reinforced rubber-like materials is related to the development of a mixed two-field displacement-pressure energy function, which leads to constitutive equations applicable for fibrereinforced materials that experience finite strains [17,18]. Obviously, this approach requires an implementationof new material model in the FE codes.

The primary objective of the present study is to develop a reliable and inexpensive model for a tyrematerial when using the available material library in LS-DYNA and to estimate the applicability ofsuch model for simulating an impact on a deformable structure. Given the complexities of the problem, viz.,large dynamic strains and contact conditions between the missile and the plate, it was decided to takethe simplest possible approach within the limitations of DYNA. Hence the second approach was used, i.e.,one-dimensional bar elements representing the reinforcing cords. The proposed discrete model for thetyre material is calibrated using the static and dynamic test data for the actual tyre material [6]. The vali-dation of the model is performed when comparing the numerical predictions with the response of a squarealuminium alloy plate to a tyre projectile impact [6] having different initial velocities and impacting theplate at 301.

2. Material characteristics

The tested tyre material has a complex structure consisting of a rubber-like material reinforced with nyloncords. The cords are placed at different angles, but in general these angles are smaller than 301 with respect tothe circumferential axis of the tyre (Fig. 1(a)). The reinforcing cords are ropes twisted from Nylon fibres andtheir stiffness in compression is significantly lower then the stiffness in tension. This material structure resultsin anisotropic properties and a non-linear behaviour when subjected to large deformations, as revealed by thetest data presented in [6]. The experimental static material properties in tension in the x and y-directions areshown in Fig. 1(b) and (c), respectively. The x-direction for the tested and modelled specimens is associatedwith the circumferential direction of an actual tyre and the y-direction corresponds to the tyre radial direction(Fig. 1(a)). The z-direction is the through thickness direction for an actual tyre. During a debris impact on astructure, large compressive deformations of the projectile and large bending deformations, and thereforeconsiderable tensile strains can occur depending on the angle of impact due to the flexibility of the projectile.Thus, to accurately model the way a tyre projectile deforms upon impact, both tension and compression dataare required.

3. Discrete model for a tyre material

The approach, which uses one-dimensional bar elements to simulate the behaviour of the reinforcing cordsin rubber, is used to model the tyre material. In general, the properties of the rubber-like part determine theflexibility of the material, whilst reinforcing cords are used to control the anisotropic properties in tension inthe x and y-direction.

It is assumed that there is no anisotropy in the z-direction but that the model material retains theorthotropic properties in tension in x and y-directions, similar to the actual material. These material propertiesare modelled using layers of cords placed at (x, y) planes, which are situated at a constant distance along thez-axis as shown in Fig. 2(a). The modelled specimens have dimensions 75� 20� 10mm3 and 20� 75� 10mm3

for tension in the x and y-direction, respectively, which are the same as the specimen dimensions from [6]. Thecords are placed at a certain constant angle with respect to the x-axis, which is determined by the uniformfinite element discretisation for the rubber part of the tyre material. It is assumed that the cords are

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Str

ess

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a)

Model

Model

Sp B

Sp D

Sp C

0

20

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60

80

0 20 40 60 80 100

Strain (75mm gauge length) (%)

Strain (75mm gauge length) (%)

Str

ess

(MP

a)

Sp ESp G

Sp F

Fibre laminates

Tread form

Z

xy

(a)

(b)

(c)

Fig. 1. Comparison in terms of nominal stress-strain between the experimental results and numerical predictions; (a) orientations with

respect to tyre, (b) tension in the x-direction (Model B), (c) tension in the y-direction (Model A).


incompressible but have linear elastic properties at tension, so that a Discrete beam formulation with Cable

material properties was used [19]. It should be noted that later versions of DYNA allow for the non linearmodelling of the cords. These cable elements are superimposed on the solid elements modelling the rubberpart by satisfying the compatibility conditions at their common nodes, using the definition associated with theLS-DYNA concept Constrained_Lagrange_in_solid [19].

The rubber part of the tyre is modelled using a Mooney–Rivlin material model [19–21] together with fullyintegrated quadratic 8-nodes solid elements with nodal rotations. It was established that this type of solidelement has better performance when large strains at the common nodes between the solid and discreteelements occur and when large gradient contact forces develop during the plate impact. No strain-rate effectsare assumed for the material model although an increase of the magnitudes of the resultant dynamic forces atcompression has been observed [6]. This increase in forces is associated with inertia effects in the tyre materialdue to the size of the specimen rather than with strain-rate sensitivity.

Due to the overlapping of the rubber material and the cords, a reduced material density for the rubber-likepart and cords is used, in order to obtain the actual density of the tyre material r0 ¼ 1020 kgm�3. It isassumed that both material components have equal density, which results in a reduced model densityr ¼ r0=ð1þ rÞ, where r ¼ V fibres=V rubber is the reinforcement fraction represented by the ratio between thevolumes of the rubber and cords components.

Due to the high ratio of the bulk modulus to the shear modulus, a refined mesh is recommended [22] inorder to overcome the numerical difficulties resulting from the incompressibility of the rubber, so that an

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Fig. 2. Models for the tensile specimens; (a) geometry of the fibre layers, (b) tension in the x-direction, (b) tension in the y-direction. Cords

are given reduced diameter for clarity.

Fig. 3. Models for the specimen compressed in the z-direction; (a) an actual cylindrical specimen, (b) an equivalent reinforced square

prism—side view, (c) square prism—top view.

D. Karagiozova, R.A.W. Mines / International Journal of Impact Engineering 34 (2007) 647–667 651

element size of 2.22� 2.27� 1.67mm3 was used when modelling the tensile material properties. Anotherreason to use small size elements in the z-direction was imposed by the possibility to model materials with highreinforcement fraction, which is typical for the tyre materials. The particular element sizes throughout theanalysis ware selected after a careful mesh sensitivity analysis on the performance of specimens’ models madeof a rubber-like material with and without reinforcement. In the former case, the mesh sensitivity analysis wasperformed for a constant angle of reinforcement.

The present approach applied to model the reinforced material, although inexpensive computationally,created problems when modelling the cylinder specimen used in compression tests [6]. In order to over-come the modelling problem and also to estimate the influence of the reinforcement on the compressiveforce, the following procedure is applied. The actual cylindrical specimen having a diameter 25mm [6]is replaced by an equivalent prism with a square base having c ¼ R

ffiffiffipp¼ 22:15 mm and height 18mm as

shown in Fig. 3. The selected solid element size is 2.46� 2.46� 1.67mm3, which determines the reinforcementangle as 18.41.

The proposed discrete model for the actual tyre material is characterised by a number of constants, namelythe Poisson ratio, nr, the material constants approximating the rubber-like material and the cords character-istics: the cross-sectional area A, Young modulus, E, Poisson ratio nc and angle of reinforcement.

ARTICLE IN PRESSD. Karagiozova, R.A.W. Mines / International Journal of Impact Engineering 34 (2007) 647–667652

4. Calibration with the static tests

The above mentioned constants, which characterise the rubber and reinforcing part of the model, areobtained by calibration using the test results for the tyre material from [6]. As discussed in [6], due to the lackof availability of separate data for the Nylon reinforcement and the rubber matrix, these properties werederived from testing the reinforced tyre rubber in various directions.

The rubber matrix of the tyre was modelled using the Mooney–Rivlin model in DYNA as described in [6].The theory given in [6] was brief, and the more general three-dimensional formulation is given here, in order tofully understand the numerical simulations and the limitations of the Mooney–Rivlin model. TheMooney–Rivlin model is derived using finite strain elasticity, which is discussed in detail by Treloar [20]and Green and Adkins [21].

In general terms, the strain energy potential, W, is given by

W ¼ A1ðI1 � 3Þ þ A2ðI2 � 3Þ þ1

D1ðJel � 1Þ2, (1)

where A1, A2, and D1 are material parameters, I1 and I2 are the first and second deviatoric strain invariantsdefined by

I1 ¼ l2

1 þ l2

2 þ l2

3 and I2 ¼ lð�2Þ

1 þ lð�2Þ

2 þ lð�2Þ

3 , (2a,b)

where the deviatoric extension ratios are given by

li ¼ J�1=3li (2c)

J is the total volume ratio, Jel is the elastic volume ratio and li are principal stretches. The initial shearmodulus and bulk modulus are given by

m0 ¼ 2ðA1 þ A2Þ and K0 ¼ 2=D1. (3a,b)

For the particular case of simple extension, the true stress t is given by [19–21]

t ¼ 2 l21 �1

l1

� �qW

qI1þ

1

lqW

qI2

� �¼ 2 l21 �

1

l1

� �A1 þ

1

lA2

� �(4)

which is consistent with Eq. (4) in [6]. Once constants A1 and A2 are obtained from uniaxial tests, three-dimensional true stress is derived using [19–21]:

tij ¼ 2ffiffiffiffiffiI3

p qW

qI3dij þ

2ffiffiffiffiffiI3p

qW

qI1þ I1

2ffiffiffiffiffiI2p

qW

qI1

� �Cij �

2ffiffiffiffiffiI2p

qW

qI1CikCjj, (5)

where d and Cij are strain tensors. It can therefore be seen that the derivation of three-dimensional stressesand strains from the model that has been calibrated using two-dimensional stress system, is complex, andincludes many assumptions. Care is required to differentiate between nominal stress and true stress. For moreinformation see [20,21]. Specifically, the Mooney–Rivlin model can become inaccurate for strains over 70%,and for cases where shear and biaxial effects dominate.

It should be noted in passing that Green and Atkins [21] consider the case of inextensible cords in a finiteelement continuum (i.e., reinforced tyre rubber). They provide analytic solutions for flexure of reinforcedcuboids and for in plane loading of reinforced membranes. Such analytic solutions could be used to furthervalidate and investigate the numerical models used here.

The static tensile material tests were simulated when a constant low velocity was applied to both ends of thespecimens and the displacements in the direction perpendicular to tension were constrained. It is evident inFig. 4 that the tensile properties of the reinforcing fibres (the reinforcing angle is 181) contribute significantlyto the response of the model to tension in the x-direction. Due to the small angle with respect to the x-axis, thecords could be subjected to considerable tensile deformations. In contrast, the reinforcing cords contributenegligibly to the tensile force in the y-direction as they are subjected mainly to rotation until considerably largetensile strains occur. In fact, the reinforcing cords do not contribute at all to the tensile force in the y-directionup to 110% strain, which is the maximum tensile strain from the test result in the y-direction (Fig. 5).

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Fig. 4. Shapes of the tensile specimen in the x-direction from simulation (Model B); (a) initial, (b) deformed at 35% strain. Cords are given

reduced diameter for clarity.

Fig. 5. Shapes of the tensile specimen in the y-direction from simulation (Model B); (a) initial, (b) deformed at 110% strain. Cords are

given reduced diameter for clarity.


A comparison between the stresses developed in the rubber part of the tyre material (recorded in the middlecross-section of the modelled specimen) and the axial stress that occurs in a single reinforcing cord is shown inFig. 6(a). One can see that the tensile material strength in the x-direction is mainly due to the reinforcement.By way of contrast, the tensile strength in the y-direction is determined only by the tensile properties of therubber material (at least for strains smaller than 110%). It is shown in Fig. 6(b) that the tensile stresses in they-direction are equal to the stresses in the rubber part of a specimen subjected to tension in the x-direction.

Fig. 7 shows compression stress strain experimental data (using a cylindrical specimen) [6] for two cases,namely (a) core only and (b) through thickness material. It was observed in the experiments that thecompressive characteristics of the core-only material (i.e. reinforced region) and the through the thickness (i.e.reinforced region and rubber outer tread) response differ considerably within the expected range of thecompressive strains and the core only material manifests larger stresses. There is a range of compressive strainsbetween 50 and 75%, approximately for the core-only material, where the stresses vary only slightly with thestrains, which can be attributed to the structural response of the specimens (e.g. local damage) but not to thematerial characteristics themselves. It is anticipated that this response is due to the physical interactionbetween the fibres in the actual material, which is difficult to model using the available elements and materialmodels in LS-DYNA. For that reason, in terms of compressive strains, it is aimed to obtain material constantsfor the model, which can predict nominal stress strain relationship closer to the one characterising thethrough-the-thickness material for strains smaller than 70–75%, as this material is closest to the characteristicsof the actual projectile. The nominal stress in the simulation is obtained when the reaction force at the edge ofthe tested specimen is divided by its initial area.

Consequently, the stress-strain relationship in tension in the y-direction (Fig. 1(c)) and the ‘through thethickness’ compression stress-strain curve (Fig. 7(b)) are used to obtain the material constants for the rubber-like material when the Poisson’s ratio is assumed as nr ¼ 0:495. The two parameters for the Mooney-Rivlinmaterial model can be obtained, e.g., either graphically or using the curve-fitting procedure implemented in

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0 5 10 15 20 25 30 35 40

Strain (%)

Str

ess

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a) Beam (axial stress)

Rubber part (von Mises stress)

00 20 40 60 80 100

5

10

15

20

25

Strain (%)

Vo

n M

ises

str

ess

(MP

a)

x-direction

y-direction

(a)

(b)

Fig. 6. Stresses resulting from quasi-static tension in the x and y-direction; (a) comparison between stresses in the rubber part and a cable

(tension in the x-direction), (b) comparison between stresses (using Model A) in the rubber parts of the tensile specimens in the x and y-

direction.


LS-DYNA. This latter procedure led to A1o0 and A240 for the Mooney-Rivlin law, when using the standardcurve-fitting procedure in LS-DYNA. In order to satisfy the Drucker postulate (ds=dl40, where s and l arethe uniaxial stress and stretch, respectively), the negative constant should be assumed equal to zero. However,the obtained stress-strain curve using only one constant resulted in a good agreement with the experimentalcurve only for small compressive strains and there was large deviation for both large compressive and largetensile strains.

Nevertheless, the constants A1 ¼ 0 and A2 ¼ 6:8Nmm�2 obtained by the standard fitting procedure for theMooney-Rivlin model are used as an initial approximation for the material model with no cords (Model A).This is required as a starting point for curve fitting, when the cords are introduced into the model. Thenominal stresses obtained from model A are compared with the tensile test data in the y-direction (Fig. 1(c))and the static compression stress (Fig. 7(b)). It should be noted that the simulation results in Fig. 7 areobtained using an equivalent cuboid specimen model as discussed previously and shown in Fig. 3.

Next, the cords are introduced (Model B). The constants calibrating the cords’ characteristics were obtainedwhen ‘fitting’ the tensile curve to the stress-strain relationship from the tensile test in the x-direction. In thiscase, the Poisson ratio for the cord material is assumed as nc ¼ 0:3 but the cross-sectional area and elasticmodulus of the cords are varied. The reinforcement in the models for the tensile tests consists of four layers ofcables (see Fig. 2). The large anisotropy in tension is achieved by increasing the axial forces in the cables andvarying the angle of reinforcement. An elastic modulus of the cords of 250MPa was assumed [6], and this gavean individual cross-sectional area A ¼ 3mm2 which results in a cord diameter of 1.95mm. Due to the linearelastic properties of the cables and the major contribution of this component to the tensile characteristics inthe x-direction, only an almost linear approximation of the tensile stress-strain relationship in this direction ispossible as evident in Fig. 1(b). The resulting volume fraction of the cords’ component is quite high asV fibre=V rubber ¼ 0:567, which determines a reduced material density as r ¼ 0:67r0, where r0 is the density ofthe actual tyre material.

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200

300

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0 20 40 60 80

20 40 60 80

100

100

Strain (%)

Str

ess

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a)

Model

BC1

BC3

0

100

200

300

400

500

0Strain (%)

Str

ess

(MP

a)

Model

BC1

BC3

(a)

(b)

Fig. 7. Comparison between the experimental results and numerical predictions (Model A) for compression in the z-direction; (a) core-

only material, (b) through-the-thickness material.

0

40

80

120

160

200

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

Strain

Str

es

s (

MP

a)

Cylinder

Prism

Fig. 8. Comparison between the static compression nominal stress-strain curves for a cylinder and an equivalent square prism (Model A).


A comparison between the static compression of a model for a non-reinforced actual cylindrical specimenand a rubber only equivalent prism is made in order to estimate the possible error when replacing the actualcylindrical specimen by a square prism (the compression of a square prism is not entirely uniaxial as it issupposed to be in the experimental set up) Fig. 8. A square prism could be used to represent the actual cylinderfor strains smaller than 70%. It should be also noted that the reinforcement of the model does not contributeconsiderably to the compression in the z-direction and only slightly increases the compressive stress due to thevariation of the Poisson effect.

The material constants for the model are obtained by minimising the overall deviation of the numericalpredictions from the material test results [6]. The smallest deviation of the predicted nominal stresses from theexperimental tensile stresses in the x-direction (Fig. 1(b)) and compressive one (Fig. 7(b)) are obtained when


using A1 ¼ 0.5Nmm�2 and A2 ¼ 4.2Nmm�2 for the Mooney–Rivlin material model and reinforcement at181 with respect to the x-axis with cords having diameter of 1.95mm and stiffness of 250MPa.

It should be noted that the numerical simulations for the calibration of the static stress-strain relationshipare done at a constant velocity of 0.5m s�1 in compression and tension due to the very significant increase ofthe computational time when decreasing the load speed. Only the tensile test in the x-direction is simulatedwith a constant velocity of 0.1m s�1 as well, but virtually no difference is observed in the response incomparison to the tension case with 0.5m s�1.

5. Verification with the dynamic tests

Although the tested material has not shown significant inertia effects during the dynamic tests [6], it isimportant to verify the proposed model with the dynamic test results due to the large flexibility of the tyreprojectile.

The dynamic verification of the model for the tyre material is done simulating impacts in the z-directionwhen using the equivalent square prism model shown in Fig. 3. The reduced density for the rubber part of thereinforced prism is r ¼ 0:737r0. The loads are applied as impacts by a mass G ¼ 12 kg with initial velocity7m s�1 and by a mass G ¼ 0.385 kg with initial velocity 50m s�1. No special surface treatment was applied tothe contact surfaces of the tested specimens, so that the measured static friction coefficient of 0.778 [6] is usedfor the simulations.

Comparisons between the numerical predictions and the test results are presented in Fig. 9(a) and (b) for alow and high impact velocity, respectively. It is evident that the selected specimen geometry, the model forthe reinforcement and the material characteristics of the tyre components lead to numerical predictions forthe dynamic load–displacement histories, which are in good agreement with the corresponding test results.A discussion of the strain rates for these cases is given in [6].

0

20

40

60

80

100

0 4 6 8 10 12 14Displacement (mm)

Lo

ad (

kN)

Model

Model

Drop Hammer BT2Drop Hammer BT1

0

20

40

60

80

100

120

0 4 6 8 10 12 14Displacement (mm)

Lo

ad (

kN)

Gas Gun BT2

Gas Gun BT3

2

2

(a)

(b)

Fig. 9. Comparison between the experimental results and numerical predictions (Model B) for the dynamic compression of a through-the-

thickness specimen (a) impact with V0 ¼ 7m s�1, G ¼ 12kg, (b) impact with V0 ¼ 50m s�1, G ¼ 0:385kg.

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0

20

40

60

80

100

120

140

0 2 4 6 8 10 12 14 16Displacement (mm)

Lo

ad (

kN)

Static

Dynamic7m/s

Dynamic50m/s

Fig. 10. Summary of the static and dynamic force-displacement histories for Model B and compression in the z-direction of a through-the-

thickness specimen.

Table 1

Characteristics of the tyre material model

Rubber part Reinforcement

Mooney–Rivlin law Cable material

A1 (N/mm2) A2 (N/mm2) nr rr (kg/m3) E (MPa) A (mm2) nc rc (kg/m

3) Angle

0.5 4.2 0.495 0.7r0–0.75r0 250 3 0.3 0.7r0–0.75r0 E181


Although no strain rate effects are taken into account, an increase of the dynamic force occurs whensimulating the dynamic tests due to the inertia effects in both rubber part and the reinforcing cables, which arerelated to the size of the material sample selected for simulation. A summary of the simulated static andthe two dynamic load-displacement histories (z-direction) in Fig. 10 gives an estimate for the dynamic effect inthe modelled specimens. It should be noted that a similar increase of the compressive load is observedexperimentally when increasing the impact velocity [6].

6. Summary of the model characteristics and model parameters

The various comparisons for the static stress-strain relationships (Figs. 1(b), (c), 7(a) and (b)) and dynamicforce–displacements histories (Fig. 9) confirm that the selected approach for modelling a reinforced rubber-like material together with the calibrated material constants is capable to reproduce the response of the actualtyre material to standard tensile and compressive tests. The constants for the material model are summarisedin Table 1.

7. Validation of the material model simulating a tyre impact on an aluminium alloy plate

7.1. Experimental arrangement [6]

In order to verify the dynamics of the tyre projectile and estimate the reliability of the selected model for thetyre material, an impact of a tyre projectile on an aluminium alloy plate is simulated. The experimentalmethod, data reduction and results are given in [6]. The numerically predicted strains in the plate, which resultfrom various impacts, are compared with the measured ones in four particular positions at the distal surface ofthe plate and particular results are presented here for impacts at 301.

The test arrangement for the tyre debris impact on a square aluminium alloy plate is described in [6]. Analuminium alloy plate 300� 300� 1.6mm3 was bolted on a thick steel frame and positioned at different angleswith respect to the axes of the projectile. The inside dimension of the frame is 260� 260mm2, which led to an


actual deformable size of the plate 260� 260� 1.6mm3. The projectile is cut from a tyre in a shape of arectangular prism (ribbon) 60mm long� 15mm wide� 30mm deep having a weight of 28 g. In allexperiments, the plate centre (x ¼ 0, y ¼ 0) coincides with the centre of the ribbon (x ¼ 0, y ¼ 0) and a ribbonprojectile is accelerated up to 136m s�1 at an (0, 0, 301) orientation in rectangular co-ordinate system (x, y, z).Strain gauges are attached at 01 and 901 to the distal side of the plate in the area of impact and at a distance100mm from the edge of the plate in the direction of impact. The records from these strain gauges [6] are usedto verify the numerically obtained strains in the x and y-directions.

7.2. Finite element model

7.2.1. Geometry, boundary and contact conditions

The structure and the material properties of the ribbon are selected according to the material modelcalibration presented in Sections 2–6 and Table 1. Eight reinforcing layers are placed parallel to the plane(x, y) at equal distances along the z axis when using 18 fully integrated solid elements for the rubber part inthis direction (Fig. 11). The reinforcement is placed at 18.41 with respect to the x-axis, which is related to theuniform discretisation (2.5� 2.5mm2) in the (x, y) plane. The elements are 1.65mm in the z-direction. Itshould be noted that the latter dimension is similar to the diameter of the cords (1.95mm).

Belytschko-Tsai shell elements 6.5� 6.5mm2 are used to model the plate where clamped boundaryconditions are assumed along all edges. The attempt to use larger shell elements with adaptive mesh increasedsignificantly the computational time and caused difficulties in the contact area for the 301 impact.

An important issue in the simulation of the rubber tyre fragment impacting an aluminium plate is the natureof the contact conditions between the two. This is complex given the large deformation in the rubber missileduring impact and the dynamic nature of the contact. However, given the complexity of the problem, it wasdecided initially to use one of the simplest contact algorithms in DYNA, viz. Nodes_to_surface_contact [19]. Itshould be noted that computational contact mechanics is a complex subject [23] and that a number ofapproaches are possible. The theory used here relates to finite deformations within the contact surface, and toa node of the contact body (slave body) sliding over several elements of the other body (master surface). Themodel assumes Coulomb friction and hence the only parameter required is the coefficient of friction(m ¼ 0:778 from [6]).

The high strain gradient in the model projectile during the initial contact imposed a condition on the timestep and it was reduced by a factor of 0.5 compared to the default time step in LS-DYNA.

7.2.2. Material properties of plate and associated strains

A piecewise linear stress-strain relationship in the true stress–true strain space is assumed for the aluminiumalloy plate according to Table 2 while the elastic modulus is E ¼ 72GPa [24]. No strain rate or plate failureeffects are taken into account in the numerical simulations. The reasons for not modelling plate rupture werethat large-scale plate rupture did not occur in experimental tests [6] and also the focus for this study was thedevelopment of a tyre rubber impact model and not a detailed model of plate behaviour.

The calculated strains in the plate are associated with the two elements on the right-hand side of the platecentre and the two elements located at 100.75mm from the plate edge in the right–hand-side half of the plate

Fig. 11. Geometry of the ribbon projectile; (a) geometry of the reinforcing layers, (b) top view. Cords are given reduced diameter for

clarity.

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Table 2

True stress-true strain relationship for the aluminium alloy [24]

Effective stress (N/mm2) 276.0 315.4 332.8 351.1 380.7 462.2 495.0

Plastic strain 0.0 0.002405 0.00426 0.00752 0.01781 0.06635 0.08906

Fig. 12. Location of the elements in the plate for the strain comparison (distal surface of the plate).


model. The locations of the elements, which are used for a comparison with the records from the strain gauges,are shown in Fig. 12. The strains for comparison in the upper elements are associated with the y-strains andcorrespond to the strain gauges SG2 (centre) and SG4 (at a distance). The elements below the x-axis are usedto compare the x-strains being associated with the strain gauges SG1 (centre) and SG3 (at a distance).

7.3. Numerical simulation of the experiment

The impacts at 301 by a ribbon projectile at an initial velocity of 135m s�1 are characterised by largebending deformations of the projectile (Fig. 13(a)) and considerable sliding along the plate, when it leaves slidemarks on the plate surface at the location of contact (Fig. 13(b)). For further information see [6].

The response of the plate and the ribbon is shown in Fig. 14 for an impact velocity of 135m s�1 at severalparticular times. It is evident, that the initial deformation phase of the projectile is characterised by bendingand sliding along the plate and this response lasts until t ¼ 0:6ms. Due to the flexibility of the ribbon, a secondcontact between the ribbon and plate occurs at t ¼ 0:68ms and again the projectile slides along the plate acertain distance before rebounding. Loss of contact occurs at 1.16ms. The two sliding contacts observed fromthe numerical simulations have their experimental evidence as slide marks on the plate as shown in Fig. 13(b) –one being associated with the area of the initial contact and another closer to the edge of the plate. Fig. 15gives a three dimensional view at 0.68 and 0.8ms, highlighting sliding behaviour. It can therefore be concludedthat the numerical simulation models the experimental behaviour from the point of view of rubber missiledeformation.

Figs. 14 and 15 show that the tyre projectile undergoes large deformations and this behaviour results in anenergy partition, which is different from the impact energy, associated with a hard projectile impact. In thepresent case of a ‘soft’ impact, a significant proportion of the initial kinetic energy is transformed intodeformation energy of the projectile (�70 J) and a relatively large energy is dissipated through the slidingcontact (�30 J). The variation of the kinetic energy of the plate (�10 J) and the projectile and the variation ofcorresponding deformation energy are compared with the sliding energy in Fig. 16(a) and (b). One can see that

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Fig. 13. Ribbon impact at 301 with V0 ¼ 135ms�1: (a) video from the impact, (b) rubber slide marks on the plate from [6].

Fig. 14. Response of an aluminium plate to a 301 impact V 0 ¼ 135m s�1 using Model B.


the sliding energy is considerably larger than the kinetic energy of the plate and is comparable to the platedeformation energy (�20 J). The largest proportion of the initial kinetic energy (�150 J) is transformed intodeformation energy of the projectile. After loss of contact, the missile continues to vibrate and energy is

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0

40

80

120

160

200

240

280

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

Time (sec)

En

erg

y (J

)

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Kinetic energy (Plate)

0

30

60

90

120

Time (sec)

En

erg

y (J

)

Deformation energy (Projectile)

Deformation energy (Plate)

Sliding energy

0 0.0005 0.001 0.0015 0.002 0.0025 0.003 0.0035

(a)

(b)

Fig. 16. Partition of the initial kinetic energy for a 301 impact with an initial velocity 135m s�1: (a) kinetic energy, (b) deformation and

sliding energy using Model B.

Fig. 15. Response of an aluminium plate to a 301 impact with an initial velocity 135m s�1 – secondary impact using Model B.


dissipated in material damping. It should be noted that the relative proportion of these energies will bedependent on missile geometry and impact conditions, and that the missile flexibility will ameliorate the energyinput into the plate.

A comparison between the experimentally recorded strain-time histories from the strain gauges SG1–SG4[6] and the corresponding numerical predictions for a ribbon impact at 135m s are shown in Fig. 17(a) and (b),respectively. The numerically predicted strain magnitudes are somewhat larger than the experimentallyrecorded ones, but this behaviour could be attributed to the highly non-homogeneous strain state at the initialimpact area. However, the overall shapes of the curves are similar.

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-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

-1 -0.5 0 0.5 1 1.5 2

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Time (msecs)

Str

ain

-0.004

-0.002

0

0.002

0.004

0.006

0.008

0.01

0.012

0.014

0.016

Str

ain

SG1

SG2

SG3

0 1 1.5 2.520.5 3

SG2

SG1

SG3

SG4

(a)

(b)

Fig. 17. Variation of the strains in four particular locations in the plate for a 301 impact with V 0 ¼ 135m s�1: (a) experimentally obtained

results (see [6]), (b) predictions from the numerical simulation using Model B.


The residual plastic strains from the simulation, which occur at the near and distal surface of the plate, areshown in Fig. 18. The largest plastic strains occur at the distal surface in the areas of the initial and secondaryimpacts. The locations of these plastic strains correspond to the slide marks observed experimentally(Fig. 13(b)). Plastic strains with smaller magnitudes occur at the near surface along the plate edge, which iscloser to the secondary impact.

7.4. Interaction between the plate and projectile depending on the loading conditions

The response of the plate to a 301 impacts with initial velocity of 135m s�1 suggests that the impact velocityand the friction coefficient are the two parameters which influence the dynamics of the projectile and thereforethe energy partition associated with the plate and ribbon deformation as well. The influence of the impact

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Fig. 18. Plastic strains in a plate resulting from a 301 impact with V0 ¼ 135m s�1: (a) near surface, (b) distal surface using Model B.


velocity and the friction coefficient on the dynamics of the tyre projectile and the residual plastic strains in theplate are briefly discussed in this section.

7.4.1. Influence of the impact velocity

The responses of the examined aluminium plate to 301 impacts with initial velocities 95, 75 and 55m s�1

were simulated and compared. The projectile tends to tumble along the plate rather than to slide whendecreasing the impact velocity but nevertheless it hits the plate twice and plastic deformations occur in bothareas of the initial and secondary impact. The location of the maximum plastic strains also changes whendecreasing the impact velocity. The largest plastic strains occur in the area of the initial impact for initialvelocities 95m s�1 and higher, while lower velocity impacts cause larger plastic strains in the area of thesecondary impact.

7.4.2. Influence of the friction coefficient

The response of the examined aluminium plate to a 301 impact with initial velocity 135m s�1 was alsosimulated and compared when assuming friction coefficients of 0.6, 0.778 (measured experimentally) and 0.95.The results from the numerical simulations show that the variation of the friction coefficient does not influencethe dynamics of the projectile for the particular impact velocity. The increase of the friction coefficienthowever causes variation of the plastic strains in the plate without affecting their locations. The largest plasticstrains (max(ep) ¼ 0.048) develop at the distal surface in the area of the initial impact for friction coefficient0.778, while the plastic strains in the area of the secondary impact are only 0.02. Comparable plastic strainsoccur in the areas of the initial and secondary impacts when assuming friction coefficients of 0.6 or 0.95.Maximum plastic strains 0.028 and 0.023 are associated with the friction coefficient 0.6, while values of 0.033


and 0.028 for the maximum plastic strains occur for the friction coefficient 0.95, but spread over a larger area.Plastic strains develop at the near surface in the area of the secondary impact as well, when assuming frictioncoefficient 0.95, which is not observed for friction coefficients 0.6 and 0.778. Presumably, the variation of theplastic strains is related to the different partition of the initial impact energy due to the large proportion of thetotal energy absorbed during the sliding contact.

8. Extension of proposed tyre model to larger tyre fragments

The validation of the material model done by simulating a tyre debris impact on an aluminium alloy plateshows that the proposed discrete model is realistic and adequate to accommodate the anisotropy andnonlinearity inherent in the tyre problem.

However, although inexpensive numerically, the proposed approach possesses two major disadvantages. Inthe real cord-rubber lamina the bonding between the surface of the cords and surface of the rubber is morethan 60% [10], whereas in the present model, the cords are attached only in the common nodes. In addition,the diameter of the cords (1.95mm) is similar to the depth of the rubber solid elements (1.67mm). The latterdoes not present numerical difficulties since the Constrained_Lagrange_in_solid concept deals only with thenode variables but nevertheless, the replacement of the necessary number of cord layers with fewer ones canchange the overall inertia properties of the projectile and consequently its flexibility under dynamic loading.A reduction of the size of the solid elements certainly leads to more accurate results.

The other disadvantage stems from the limited element dimensions that can be accommodated since themesh arrangement is not arbitrary but must be related to the cord angles.

The above limitations of the discrete model can present difficulties when used for large tyre projectiles.Having in mind the described features of the material model, a limited increase of the element size can berecommended. Once the size of the solid elements is decided a scaling coefficient for a coarse mesh, b, can beobtained for the cable material characteristics. For larger solid elements than the ones used in the model forthe small projectile (while preserving the reinforcing angle of approximately 181):

b ¼ ðdy=f zÞcoarse mesh=ðdy=f zÞfine model mesh40 (6)

in order to maintain the reinforcement fraction characteristic of the coarse mesh the same as for therefined material model. In Eq. (6), dy is the distance between the reinforcing cords in the y-direction andf z is the number of cord layers per unit thickness (z-direction) associated with the ‘coarse mesh’ and therefined mesh of the material model proposed in this paper. For the calibrated material model

Fig. 19. Finite element mesh for full-scale tyre fragment simulation. Cords are given reduced diameter for clarity.


ðdy=f zÞfine model mesh ¼ 9:365 mm2. The cord diameter is then scaled as the square root of b. This approach wasused to model a full scale, industry standard, tyre fragment test [24]. A tyre fragment, of dimensions425� 100� 28mm3 and mass 1.335 kg (wasted in the central section) is folded and placed in a large gas gun.When the fragment exits the gun barrel, it un-wraps and impacts the aircraft structure at a velocity of

Fig. 20. Comparison between experiment [25] and simulation [26] for full-scale wing access panel test: impact velocity 110m s�1 and

impact angle 301.


114m s�1 and in the orientation required. The aircraft structure of interest here was a wing access panel[24,25]. Fig. 19 shows the tyre fragment mesh, where each element is 7.46� 7.2� 3.5mm3, which should becompared to the previous case of 2.5� 2.5� 1.67mm3 (Fig. 11). For the selected coarse mesh, thereinforcement angle is 17.71 and ðdy=f zÞcoarse mesh ¼ 28:572 mm2 taking into account that seven layers of cordsare placed inside the projectile. Thus, b ¼ 3:047 for the current discretisation. In order to maintain thestrength characteristic of the reinforcing layer unchanged, the cross-sectional area of the cords is scaled bycoefficient b , which results in a cord diameter of 3.4mm while the elastic modulus for the cable materialremains the same.

Fig. 20 compares the simulation [26] with experimental test [25] for an impact angle of 301. It can be seenthat there is good comparison in respect of gross deformation fields. Ref. [26] also discusses strains at variouspositions on the wing access cover, and shows good agreement between experiment and simulation. It shouldbe noted that the full-scale model was demanding computationally, taking over a day on a large Sun server.

9. Concluding remarks

A numerical model for tyre fragment impact has been developed using standard material models in DYNAand using an approximation for the fibre reinforcement. The tyre model is approximate given the necessity forcomputationally robust and efficient modelling for industrially relevant problems. The model has beenvalidated for small scale tyre fragment tests and for full-scale tyre fragment tests. Therefore, it can beconcluded that the proposed simulation approach is accurate for a wide variety tyre impact scenarios,although demanding in computing time.

It should be noted that there are limitations to the Mooney–Rivlin rubber model used here, and that thereare more accurate rubber models in DYNA, which, however require a larger number of constants. Also, thereare more sophisticated elastomeric models in the literature, see for example [27]. In addition, it is notstraightforward to obtain the stresses within the tyre fragment directly using the output variables inLS-DYNA and the present simulation says nothing about the possible debonding between the rubber matrixand the reinforcing fibres. In fact, the diameter of the cords is comparable to the depth of the rubber elementsand although approximates correctly the anisotropic material properties, it does not even closely approximatethe physical connection between the cords and the rubber in the actual material. If this was required, thenindividual fibres would have to be modelled or some equivalent damage theory developed (e.g. see [28]).

However, all this sophistication would increase computational demand and may lead to numericalinstabilities, e.g., strain softening. It is proposed that, in the short term, a more productive avenue would be tosimplify the current model by increasing mesh size or using simpler finite elements, to reduce computationaltime for real problems.

As far as the contact algorithm is concerned, the standard contact model in DYNA gives good qualityanswers and tyre fragment behaviour seems to be insensitive to friction conditions. An adaptive mesh for theplate would seem to be unnecessary.

Acknowledgement

The support through EU Framework 5, DG XII – Competitive and sustainable growth—Key Action 4—New perspectives in Aeronautics: Crashworthiness under high velocity impact (CRAHVI) Contractno. G4RD-CT-2000-00395, is greatly acknowledged. Thanks are also due to Martin Kracht and ChristianBergler, of Cadfem Gmbh, for help in the full scale simulations and Jean Philippe Gallard, of CEAT, for fullscale testing data.

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dx.doi.org/10.1016/j.ijimpeng.2006.02.005

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